Let $\lambda$ be Lebesgue measure and $\mu$ be counting measure both regarded as Borel measures on $I=\left[0,1\right] $. Let $\Delta$ be the diagonal in $I\times I:\Delta=\left\{\left(x,y\right) | x=y\right\} $. Let $f$ be a characteristic function of $\Delta$.
I don't know how to compute $\int_{I\times I}fd\lambda \times d\mu$. This is part of a question:
$(a)$ Compute the integrals:$\int_{I}\left(\int_{I}fd\lambda\right)d\mu$ , $\int_{I}\left(\int_{I}fd\mu\right)d\lambda$ and $\int_{I\times I}fd\lambda \times d\mu$
$(b)$ Why the result above does contradict Fubini theorem.
From the definition I know that:$\int_{I}\left(\int_{I}fd\lambda\right)d\mu$=$0$ and $\int_{I}\left(\int_{I}fd\mu\right)d\lambda$=$1$. And since counting measure is not $\sigma$-finite which is why two results above we've calculated does not contradict Fubini's Theorem.
